Search for the Fundamental QCD String in AdS Space

1
Search for the Fundamental QCD String in AdS
Space
Arizona--- July 28, 2006
Richard C. Brower Boston University
2
Basic QuestionAre QCD flux tubes equivalent to
zero width fundamental strings in curved space?
If so what is there world sheet action?
31 Yang Mills
32 AdS5 like String Theory
?
?
?
?
(Leinweber, Univ of Adelaide)
(S. Brodsky, SLAC)
3
Outline

• I. Fundamental vs Phenomenological String Duals
• II. Maldacenas AdS/CFT dictionary
• III. Limits Regge/DIS/pp --- general
  consequences

• Classics example sine-Gordon massive
  Thirring Model
• Expansion parameter for QDC string before
  QCD!
• Success and Failures of flat space dual QCD
  string

• Conformal symmetry --- strings w.o.
  confinement! (Liouville mode)
• Stretch string, conformal spectrum,
  confinement, Luscher term
• Confinement deformations and the IR cut-off
  --- glueballs

• -- Strong Coupling BFKL, SL(2,C), OPEs, Spin
  Chains
• -- Flavor and bottom up approach (DW déjà vu,
  2-d tHooft solution)
• Lattice computation QCD string N 1 data!
• Putting gravity back in? Einstein vs Chiral
  Lagrangian

4
start
Hadronic Boostrap, zero width approx(1965)
FESR/Regge
Good IR
) Discovery of String Theory(1968)
Discovery of QCD/Yang Mills(1973)
No ghosts/No Tachyon
Lattice QCD (1974)
Top Down
d91 SUSY/gravity
Anomaly Cancellation
IIA
IIB
I
1/Nc tHooft expansion at week and strong (1974)
HO
HE
Non- perturbative D-branes
Landscape Swamp!
Maldacena YMAdS Strings(1997)
Wilson Loop Eq Matrix Models
AdS/CFT Strings w.o. Confinement -- Good UV
Break Conformal/SUSY
Confining QCD-like Duals
Qualitative Results Glueballs,
confinement,Tc,.. General consequences entropy
bound, Regge,..
String Spectra from Lattice
Oasis
Bottom Up
5
Apologies for neglect of important bottom up
efforts

• String Excitations between Static Sources
• Juge, Kuti, Morningstar
• Luscher, Weisz
• H. Meyer
• Glueballs at Large Nc
• Morningstar,Peardon
• Teper, F. Bursa, B. Bringolz
• Large Nc Matrix Model Reduction
• Narayanan, Neuberger, Kiskis,
• Holographic Model of Hadrons
• Karch, Katz, Dam, Son, Stephanov
• ETC

QCD string abstract) hep-lat 204 Spires t QCD
String ) 241 Spires t AdS ) 1710 !
6
Example of Exactly Dual Quantum Theories

• sine-Gordon Theory massive Thirring Model.
• expansions g2 1/g2
• elementary field solitonic
  bound state

S. Coleman Phys. Rev. D 11, 20882097 (1975)
S. Mandelstam Phys.Rev.D113026,1975 Soliton
operators for the quantized sine-Gordon
equation Operators for the creation and
annihilation of quantum sine-Gordon solitons are
constructed. The operators satisfy the
anticommutation relations and field equations of
the massive Thirring model. The results of
Coleman are thus reestablished without the use of
perturbation theory. It is hoped that the method
is more generally applicable to a
quantum-mechanical treatment of extended
solutions of field theories.
7
Extreme Limits of QCD
Different String Theories (vacua) are Dual to
each other.
String/Gravity
N 1, nf 1
(tHooft)
N 0
(Armoni,Shifman,Venziano)
Asymptotically Free (UV/Short Distances)
Flux Tubes/Spectra (IR/Long Distances)
Ncolor
QCD
g2
1/g2

?B
kT
Color Supercond (Dense quarks)
Chiral Restored (High Temp)
8
First attempt to construct a QCD String

• 1968 (before QCD!)
• Identify small parameter ?hadron/Mhadron ' 1/10
  
• Hadronic Bootstrap) Discovery of String
  Perturbation Theory.
• 1974 (after QCD) tHooft 1/Nc expansion of
  SU(Nc) QCD
• Both Weak (Feynman) and Strong (lattice)
  expansions give

?Euler Vertices - Edges Faces 2 - 2 Handle
(glueballs) - Boundaries (quarks)
9
Regge trajectory
?(t) ' ? t ?0 where ?(t M2) J
10
Small parameter of zero width approx ?/M ' 0.1
Parity partners MacDowell symmetry
11
Dolan-Horn-Schmid duality ---Phys.Rev. 166, 1768
(1968)
t
t
?-
?
?
?
n

??
s
s
?-
?-
?
?-
t-channel Regge amplitude A ' (-s)a(t) smoothly
interpolates s-channel resonances
Popular Regge Parameterization (1965)
12
Veneziano Dual Pion Amplitudey
y Neveu-Schwarz Quark model of dual pions, 1971
??(0) 1/2 implies Adler zeroA(s,t) ! 0, as p1
! 0.
The open QCD string (pion amplitudes) at low
energies are equivalent to the Chiral Lagrangian
at Nc 1
13
Unitarity Open (Meson) ) Closed
(Glueball/Pomeron) String

s !
? B 1 ) O(1/N)
? H 1 ) O(1/N2)
t channel Two meson exchange
t channel Single glueball exchange
t
t
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Failures of (flat space) Strings for QCD

• (i) ZERO MASS STATES 1--
  gauge/ 2 graviton
• (ii) EXTRA SYMMETRY SUSY
• (iii) EXTRA DIMENSION 91
  10
• (iv) NO HARD PROCESSES! totally
  wrong dynamics

Stringy Rutherford Experiment
At WIDE ANGLE s,-t,-u gtgt 1/a
Strings are too soft
Form Factors do not exist!
15
Need to higgs the Graviton/Photon into a
massive Glueball/Rho
J
2
Glueball/Pomeron
Graviton
1--
Rho
Photon
Gluon
M2
t0
tgt0
tlt0
Maldacena Solution put 10-d (super) strings in
curved space
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Open (Gluons) Close (Graviton) duality in near D
brane Horizon
AdS5 S5 string N 4 Super Conformal YM
in 4-d

• 3-branes (13 world volume) -- Source for open
  strings and closed strings

Their mass curves the space into AdS5 emits
closed string (graviton)
Dynamics of N D3 branes at low energies is
(Super) SU(N) YM.
gmn gravitons
Am gluons
D3-branes
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Maldacenas AdS/CFT Dictionary

• SO(2,4) conformal symmetry in 4-d is mapped into
  the isometries of 5-d Anti-de Sitter and extra R
  symmetries on a 5-d Sphere.
• The AdS5 space is hyperbolic
• At weak coupling the space is highly curved
• All operators in CFT correspond to string states
  (fields) in AdS with
• String/Gauge Duality without confinement!

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Repeat

• AdS5 S5 string N 4 Super Conformal YM
  in 4-d
• IS ALMOST CERTAINLY AN EXACT STRING/GAUGE
  DUALITY.
• IF SO CONTRARY TO CONVENTIONAL WISDOM EXACT
  STRING/GAUGE DUALITY DOES NOT REQUIRE CONFINING
  FLUX TUBES!

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Isometries of AdS5 S5
Near Horizon geometry is a product of a
Hyperbolic space and a 5 sphere
with z R2/r
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AdS5 with an IR cut-off
Large Sizes
Add Confinement IR wall!
String/Glueball
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Non-zero QCD string tension and Mass Gap
with QQ potential V(L ) const/ L
Conformal Background
Witten PS KS MN
give QQ potential V(L ) a'qcd L c/L L,
Background with IR cut-off
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Lattice QCD4 Stretched String Spectrum
Juge, Kuti, Morningstar
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Transverse String excitations
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(No Transcript)
25
Excited states (Semi-classical limit)
2 Transverse (Goldstone) Modes
Radial (longitudinal) Mode
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String Level structure
See Kuti
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Profile of Wave Functions in Cut-off AdS5 Model
?(r)
IR WALL
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Lattice Data vs AdS Confining Gauge Theory at ?
1
Lattice QCD Glueball Spectrum By Morningstar and
Peardon
Mass Gap
Brower, Shamir, Tan
Moringstar, Peardon
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IIA Classification of QCD4
States from 11-d GMN
States from 11-d AMNL
Subscripts to JPC refer to Pt -1 states
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Hard Component in High Energy Scattering

• Power behavior at wide angle (dimensional/parton
  counting rules)
• BFKL hard diffractive scattering at strong
  coupling

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Intuitive Approach Soft vs Hard in ADS QCD
(Polchinski Strassler RCB C-I Tan
hep-th/0207144)

• Red Shift
• Proper Length ? s (r/R) ?
  x
• Local Momentum plocal? (R/r) p?
  (large p in IR!)

• Wide angles obeys power counting rules
  (Polchinki Strassler)
• Astring( ? R2 s/r2, ? R2 t/r2) exp - R2 s
  log(s) /r2
• Dominant piece is conformal scaling for r ! 1

• Regge region is an average for r

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• Soft IR region r ' rmin, gives Regge pole
  with slope a'qcd a' R3/r3min

• The shrinkage'' is caused the soft stringy
  form factor'' in impact parameter

• Hard IR region BFKL-like Pomeron with almost
  flat cut in the j-plane

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Ultra local Model in AdS5
IR
UV
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BFKL Balitsky Lipatov Fadin,Kuraev,Lipatov 75
Diffusion in virtuality k?

• QCD perturbation theory 1st order in ?s and all
  orders (?s logs)n
• Implies planar diagrams (e.g. Nc 1) and
  conformal scaling
• BFKL is essentially a large Nc CFT results!

35
Diffusion in log(k?) is familiar in Regge but ...!
p1
p3
s (p1p2)2 ' m1 m2 expy
t (p1p3)2 -q2?
p4
p2
Take Fourier transform
How do we combine diffusion in x? and log(k?) ?
36
Strong Coupling YM is computed in String Theory
(Brower, Polchinski, Strassler and Tan
hep-th/0603115)
Strong Coupling
at t 0
Diffusion in warped co-ordinate
Compare with weak Coupling
37
Main Lesson from AdS/CFT dual description of
Diffraction
38
Hard versus Soft Diffraction (Lightcone
Derivation)
With X ?
39
The Schwarz-Christoffel trans maps the upper
half plane (a) into the light-cone strip ? i?
(b)
40
Reduction to 1-d Path Integral
where
41
Regge Behavior is diffusion for time log(s) in
impact parameter space (and AdS radial space)
Rapidity y log(s/s0) and t - q2?
Boosts increases size of hadronic string
exp - ? q2? log(s) ? exp - b2/(? log(s))
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AdS5 Modifications
43
Strong Coupling Pomeron

• V(u) -t e-u 0 lt u lt 1
• Attractive for t gt0, Regge Pole
• BKLF cut
• t lt 0 only scattering state for BKLF

V(u)
t lt 0
u
t gt0
Hard Wall at r r_min
44
Hardwall Regge Spectrum and Cut
45
First and Second Sheet
46
V running
47
(Strong) Running Coupling
48
N 4 Strong vs Weak BFKL
weak 1st
Strong
weak 2nd
49
Boosts AdS5 ! AdS3 AdS3 ?
BFKL/DGLAP
AdS3 dual
Regge/DIS
Boost
Boost
AdS5 dual
String/Gravity
Yang Mills/Weak
DIS Collinear subgroup SLR(2,R) SLL(2,R) x
x0 x3 BFKL Transverse subgroup SL(2,C) z
x1 i x2 Share Dilatation operator

50
All coupling form ?(j) DGLAP vs BFKL
(4,2) and (0,2) have zero anomalous dimension
? 0, BFLKL
? 0 DGLAP
inversion symmetry ? ? 4 - ?
51
On shell Reggeon vertex operator
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Pomeron Vertex Operator
where
53
QCD and a Holographic Model of Hadrons Erlich,
Katz, Son, Stephanov, hep-ph/05011
Mesons A generalized weak coupling (chiral
theory) 5-d theory
54
Holographic Picture is almost identical to DW
fermions
s 1
s 2
s M
s L_s
qL
QL
QR
qR
RIGHT
LEFT
qR
QR
QL
qL
55
5-d Vector Current ? 4-d Vector/Axial Current
Vector
Axial
56
Conclusions

• Maldacenas AdS/CFT conjecture
• String/Gauge duals exist and are natural
• Qualitative failures of the old flat space QCD
  string are removed.
• It is now possible to compute in strong coupling
  confining theories
• Seek model impendent feature in this limit.
• But the detail form of the QCD string is not
  known!
• Even the d.o.f on the string are not identified.
• The lattice can give definite spectral data for
  the QCD string
• keep the enterprise honest
• guide the search for the holographic dual to
  Yang Mills QCD
• The discovery of the QCD string would put gravity
  and Yang Mills theory into the same framework.